Certain properties of the enhanced power graph associated with a finite group

TL;DR

This paper investigates the properties of enhanced power graphs of finite groups, including connectivity, regularity, and bounds on the Wiener index, with a focus on nilpotent groups and their structural characteristics.

Contribution

It classifies groups based on graph properties like minimum degree, connectivity, and regularity, and provides bounds for the Wiener index in nilpotent groups.

Findings

Identified groups where minimum degree equals vertex connectivity.

Classified groups with strongly regular enhanced power graphs.

Established bounds for the Wiener index in nilpotent groups.

Abstract

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain a lower bound and an upper bound for the Wiener index of $\mathcal{P}_E(G)$, where $G$ is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.